Methods of Election. 231 



Incomplete Voting Papers. 



There is a point of some practical importance to be con- 

 sidered in connection with the proposed method. If the 

 number of candidates was large, some of the electors might 

 not be able to make out a complete list of the candidates in 

 order of preference. We have then to consider how voting 

 papers, on which the names are not all marked in order of 

 preference, are to be dealt with. Such a voting paper may 

 be called incomplete. In order to examine this question, 

 let us first suppose, for the sake of simplicity, that there are 

 only three candidates A, B, C, and that the votes tendered 

 are of one of the forms AB, BA, C, that is to say, that all 

 the electors who put A first put B second, that all who put 

 B first put A second, and that all who vote for C mark no 

 second name. In accordance with the proposed method, for 

 each paper of the form AB, two votes would be given to A 

 and one to B ; and for each paper of the form BA, two votes 

 would be given to B and one to A. The question arises, 

 however : is a paper of the form C, that is, a plumper for C, 

 to be counted as one vote or as two votes for C ? If it be 

 counted as one vote only, it is clear that C might be defeated 

 even if he had an absolute majority of first votes in his 

 favour. For if we suppose AB=BA=a, and G=c, it is clear 

 that the scores of A and B will each be equal to 3a, and 

 that of C to c. Thus C will be defeated unless c > 3a ; but 

 if c > 2a, there is an absolute majority for C. Hence, then, 

 we may be led into error if each plumper for C be counted 

 as one vote only. If, on the other hand, a plumper be 

 counted as two votes, it is clear that C might win even if 

 there were an absolute majority against him. For the score 

 of C will now be 2c, and C will win if 2c > 3a. But if 

 2c < 4<x, there is an absolute majority against C. Thus we 

 should also be led into error if each plumper be counted as 

 two votes. If, however, we agree to count a plumper as 

 three halves of a vote, neither of these errors could occur. 

 This course is readily seen to be the proper one in any case 

 of three candidates, for it clearly amounts to assuming that 

 the electors who plump for C are equally divided as to the 

 merits of A and B. For if a 1 , b l , & denote the numbers of 

 plumpers for A, B, C respectively, and if we agree to con- 

 sider all the electors who plump for A as being equally 

 divided as to the merits of B and C, the effect of the a l 

 plumpers for A would be to give 2 a x votes to A, and \ a 1 each 



